In the U.S, the bit is equal to 1/8 of a dollar or 12.5 cents. In the U.S., the "bit" as a designation for money dates from the colonial period, when the most common unit of currency used was the Spanish dollar, also known as "piece of eight", which was worth 8 Spanish silver reales. One eighth of a dollar or one silver real was one "bit".

With the adoption of the decimal U.S. currency in 1794, there was no longer a coin worth of a dollar but "two bits" remained in the bit and 15¢ a long bit.

Robert Louis Stevenson describes his experience with bits in Across the Plains, p. 144:1In the Pacific States they have made a bolder push for complexity, and settle their affairs by a coin that no longer exists – the BIT, or old Mexican real. The supposed value of the bit is twelve and a half cents, eight to the dollar. When it comes to two bits, the quarter-dollar stands for the required amount. But how about an odd bit? The nearest coin to it is a dime, which is, short by a fifth. That, then, is called a SHORT bit. If you have one, you lay it triumphantly down, and save two and a half cents. But if you have not, and lay down a quarter, the bar-keeper or shopman calmly tenders you a dime by way of change; and thus you have paid what is called a LONG BIT, and lost two and a half cents, or even, by comparison with a short bit, five cents.

"Put my two cents in" originates from the older "put my two bits in" and has its origin in the game of poker. When playing poker you have to make a small bet before the cards are dealt called an "ante" to begin play in that hand.
This phrase draws an analogy to the poker ante (two bits) and gains your entry into the conversation.

Two bits means one quarter (currently the American twenty five cent piece). This comes from the older term "piece of eight".

Today we have coins minted in different denominations - nickel, dime, and quarter in the U.S. - but this was not always so. Gold and silver coins once served as currency, with the value of the coin equal to the value of the gold or silver contained in the coin. To obtain currency valued at less than a full gold coin, coins would be scored and split into pieces. This is how one would make change so to speak.

Coins could be split into halfs, quarters, and eighths. One eighth of a coin was called a "piece of eight" and also called a "bit". Two pieces of eight is equal to one quarter. Hence "two bits" is a quarter.

Person 1: "....That is just my two cents"

Person 2: "You mean, 'My Two Bits' you moron... by misquoting the phrase, you illustrate just how stupid you are and your opinion should be disregarded."

Deriving from the "Shave and Hair cut, two bits" jingle, "Two bits" refers to a woman who does not shave/wax/remove body hair on legs, arm pits, facial hair, etc. The words two bits are used alone to reference the jingle with the implication that a shave/wax/haior removal is in order.

(At the beach)
Man 1: "Look at the hair on her thighs"
Man 2: "Yeah, two bits allright"